Open Access

Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions

Advances in Difference Equations20112011:109570

Received: 4 December 2010

Accepted: 11 February 2011

Published: 9 March 2011


We discuss conditions for the existence of at least one solution of a discontinuous parabolic equation with lower semicontinuous right hand side and a nonlocal initial condition of integral type. Our technique is based on fixed point theorems for multivalued maps.

1. Introduction

Let be an open bounded domain in , , with a smooth boundary . We denote the norm (usually the Euclidean norm) of by . Let be a positive real number. Set and . For we denote its partial derivatives (when they exist) by .

Let denote the Banach space of continuous functions , endowed with the norm
For , we say that is in if is measurable and , in which case we define its norm by
Consider the linear nonhomogeneous problem
with the following nonlocal initial condition:
Here, is an elliptic operator given by
We will assume throughout this paper that the functions are Hölder continuous, , and moreover, there exist positive numbers such that
Let be continuous. For the problem (1.3), (1.4) together with initial condition

we have the following classical result.

Lemma 1.1 (see [14]).

Assume that the function is Hölder continuous on and is continuous on . Then problem (1.3), (1.4), (1.8) has a unique solution , which for each , is given by

where , is the Green's function corresponding to the linear homogeneous problem. This function has the following properties (see [1, 4]).
  1. (i)


  2. (ii)


  3. (iii)


  4. (iv)

    for .

  5. (v)

    are continuous functions of .

    In addition to the above, satisfies the following important estimate.

  6. (vi)

    , for some positive constants (see [2]).


Since , it is clear that the functions   and are continuous. Let and let Also, property (vi) above shows that .

In this paper, we consider a nonlocal problem for a class of nonlinear parabolic equations with a lower semicontinuous multivalued right hand side. More specifically, we consider the following problem,

Parabolic problems with discontinuous nonlinearities arise as simplified models in the description of porous medium combustion [5], chemical reactor theory [6]. Also, best response dynamics arising in game theory can be modeled by a parabolic equation with a discontinuous right hand side [7, 8]. Parabolic problems with discontinuous nonlinearities have been also investigated in the papers [913]. On the other hand, parabolic problems with integral boundary conditions appear in the modeling of concrete problems, such as heat conduction [14, 15] and thermoelasticity [16]. Also, the importance of nonlocal conditions and their applications in different field has been discussed in [17, 18]. Several papers have been devoted to the study of parabolic problems with integral conditions [19, 20]. Next, we state some important facts about multivalued functions and results that will be used in the remainder of the paper.

A subset is measurable if belongs to the -algebra generated by all sets of the form where is Lebesgue measurable in and is Borel measurable in . Let and be Banach spaces. denotes the set of all nonempty subsets of . The domain of a multivalued map is the set Dom( has closed values if is a closed subset of for each and we write . Also, denotes the set of all nonempty closed and convex subsets of . is bounded if is called lower semicontinuous (lsc) on if is open in whenever is open in , or the set is closed in whenever is closed in . For more details on multivalued maps, we refer the interested reader to the books [2124].

Let denote the Kuratowski measure of noncompactness. See [25] for definitions and details.

Theorem 1.2 (see [26, Theorem 3.1]).

Let be a separable Banach space. Assume the following conditions hold. There exists , independent of , with for any solution to   a.e. on for each is a closed map, is a bounded subset of , and for all with strict inequality if . Then the inclusion has a solution .

2. Main Result

By a solution of problem (1.10), (7), (8) we mean a function such that there exists a function with for each and (1.3), (1.4), (1.5) hold.

Theorem 2.1.

Assume that the following conditions are satisfied.

(HF) is measurable, is lsc for a.e. , there exist such that with 2Vol and there exists such that for any bounded set ,

(Hk) is continuous, bounded and there exists such that .

Then problem (1.10), (7), (8) has a solution provided that   .


We shall follow the ideas developed in [27]. It follows from the integral representation (1.9) that any solution of (1.10), (7), (8) is a solution of the operator inclusion
for , where
where is given by
while is given by
First, we show that solutions of (2.1) are a priori bounded. We have
where , that is for each . Since is bounded there exists such that . It follows from the properties of the Green's function and the assumption (HF) that
Equation (2.7) implies that
Therefore, there exists , independent of , but depending on and the Green's function such that any possible solution of (2.1) satisfies

Let . Then is nonempty, closed, and bounded subset of .

Since the multifunction has nonempty, closed and convex values, it follows that has nonempty, closed, and convex values. Since is a continuous single valued operator, it is clear that has nonempty, closed, and convex values. Next, we can easily show that is a closed map (i.e., has a closed graph) and is a bounded subset of .

Finally, we show that for any bounded subset . So, let . Then, since , we have
It follows from the assumption that

This shows that is a condensing multivalued map.

By Theorem 3.1 in [26], has a fixed point in , which is a solution of problem (1.10), (7), (8). This completes the proof of the main result.



This work is part of an ongoing research project FT090001. The author is grateful to KFUPM for its constant support. The author would like to thank an anonymous referee for his/her comments.

Authors’ Affiliations

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals


  1. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, USA; 1964:xiv+347.Google Scholar
  2. Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow, Russia; 1967. English translation: American Mathematical Society, Providence, RI, USA, 1968Google Scholar
  3. Lieberman GM: Second Order Parabolic Di¤erential Equations. World Scientific, River Edge, NJ, USA; 1996:xii+439.View ArticleGoogle Scholar
  4. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.MATHGoogle Scholar
  5. Feireisl E, Norbury J: Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities. Proceedings of the Royal Society of Edinburgh A 1991,119(1-2):1-17. 10.1017/S0308210500028262MathSciNetView ArticleMATHGoogle Scholar
  6. Fleishman BA, Mahar TJ: A step-function model in chemical reactor theory: multiplicity and stability of solutions. Nonlinear Analysis 1981,5(6):645-654. 10.1016/0362-546X(81)90080-8MathSciNetView ArticleMATHGoogle Scholar
  7. Deguchi H: On weak solutions of parabolic initial value problems with discontinuous nonlinearities. Nonlinear Analysis, Theory, Methods and Applications 2005,63(5–7):e1107-e1117.View ArticleMATHGoogle Scholar
  8. Hofbauer J, Simon PL:An existence theorem for parabolic equations on with discontinuous nonlinearity. Electronic Journal of Qualitative Theory of Differential Equations 2001, (8):-9.Google Scholar
  9. Cardinali T, Fiacca A, Papageorgiou NS: Extremal solutions for nonlinear parabolic problems with discontinuities. Monatshefte für Mathematik 1997,124(2):119-131. 10.1007/BF01300615MathSciNetView ArticleMATHGoogle Scholar
  10. Carl S, Grossmann Ch, Pao CV: Existence and monotone iterations for parabolic differential inclusions. Communications on Applied Nonlinear Analysis 1996,3(1):1-24.MathSciNetMATHGoogle Scholar
  11. Carl S, Heikkilä S: On a parabolic boundary value problem with discontinuous nonlinearity. Nonlinear Analysis: Theory, Methods & Applications 1990,15(11):1091-1095. 10.1016/0362-546X(90)90156-BMathSciNetView ArticleMATHGoogle Scholar
  12. Pisani R: Problemi al contorno per operatori parabolici con non linearita discontinua. Rendiconti dell'Istituto di Matematica dell'Università di Trieste 1982, 14: 85-98.MathSciNetGoogle Scholar
  13. Rauch J: Discontinuous semilinear differential equations and multiple valued maps. Proceedings of the American Mathematical Society 1977,64(2):277-282. 10.1090/S0002-9939-1977-0442453-6MathSciNetView ArticleMATHGoogle Scholar
  14. Cannon JR: The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics 1963, 21: 155-160.MathSciNetMATHGoogle Scholar
  15. Ionkin NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differential Equations 1977, 13: 204-211.MATHGoogle Scholar
  16. Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983,40(4):468-475.MathSciNetMATHGoogle Scholar
  17. Balachandran K, Uchiyama K: Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces. Proceedings of the Indian Academy of Sciences 2000,110(2):225-232. 10.1007/BF02829493MathSciNetMATHGoogle Scholar
  18. Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Journal of Mathematical Analysis and Applications 1991,162(2):494-505. 10.1016/0022-247X(91)90164-UMathSciNetView ArticleMATHGoogle Scholar
  19. Dai D-Q, Huang Y: Remarks on a semilinear heat equation with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2007,67(2):468-475. 10.1016/ ArticleMATHGoogle Scholar
  20. Olmstead WE, Roberts CA: The one-dimensional heat equation with a nonlocal initial condition. Applied Mathematics Letters 1997,10(3):89-94. 10.1016/S0893-9659(97)00041-4MathSciNetView ArticleMATHGoogle Scholar
  21. Aubin J-P, Cellina A: Differential Inclusions. Springer, Berlin, Germany; 1984:xiii+342.View ArticleGoogle Scholar
  22. Aubin J-P, Frankowska H: Set-Valued Analysis, Systems & Control: Foundations & Applications. Volume 2. Birkhäuser, Boston, Mass, USA; 1990:xx+461.MATHGoogle Scholar
  23. Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.Google Scholar
  24. Hu S, Papageorgiou NS: Handbook of Multivalued Analysis, Vol. I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000.View ArticleGoogle Scholar
  25. Kamenskii M, Obukhovskii V, Zecca P: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications. Volume 7. Walter de Gruyter, Berlin, Germany; 2001:xii+231.MATHGoogle Scholar
  26. Agarwal RP, O'Regan D: Existence criteria for operator inclusions in abstract spaces. Journal of Computational and Applied Mathematics 2000,113(1-2):183-193. 10.1016/S0377-0427(99)00252-6MathSciNetView ArticleMATHGoogle Scholar
  27. Byszewski L, Papageorgiou NS: An application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem. Journal of Applied Mathematics and Stochastic Analysis 1999,12(2):179-190. 10.1155/S1048953399000180MathSciNetView ArticleMATHGoogle Scholar


© Abdelkader Boucherif. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.